Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 8 (2012), 040, 16 pages
The Vertex Algebra M (1)+
and Certain Af f ine Vertex Algebras of Level −1
Dražen ADAMOVIĆ and Ozren PERŠE
Faculty of Science, Department of Mathematics, University of Zagreb,
Bijenička cesta 30, 10000 Zagreb, Croatia
E-mail: adamovic@math.hr, perse@math.hr
URL: http://web.math.pmf.unizg.hr/~adamovic/
Received March 09, 2012, in final form July 01, 2012; Published online July 08, 2012
http://dx.doi.org/10.3842/SIGMA.2012.040
Abstract. We give a coset realization of the vertex operator algebra M (1)+ with central
charge `. We realize M (1)+ as a commutant of certain affine vertex algebras of level −1
in the vertex algebra LC (1) (− 12 Λ0 ) ⊗ LC (1) (− 12 Λ0 ). We show that the simple vertex algebra
`
`
LC (1) (−Λ0 ) can be (conformally) embedded into LA(1) (−Λ0 ) and find the corresponding
`
2`−1
decomposition. We also study certain coset subalgebras inside LC (1) (−Λ0 ).
`
Key words: vertex operator algebra; affine Kac–Moody algebra; coset vertex algebra; conformal embedding; W-algebra
2010 Mathematics Subject Classification: 17B69; 17B67; 17B68; 81R10
1
Introduction
In the last few years various types of W-algebras have been studied in the framework of vertex
operator algebras (see [3, 4, 9, 13, 26]). In this paper we will be focused on W-algebras which
admit coset realization. To any vertex algebra V and its subalgebra U , one can associate a new
vertex algebra
Com(U, V ) = {v ∈ V | un v = 0 for all u ∈ U, n ≥ 0}
called the commutant (or coset) of U in V . This is a very important construction in the theory of
vertex operator algebras, because it gives a realization of a large family of W-algebras. Another
important construction is the orbifold construction, where a new vertex operator algebra is
obtained as invariants in a given vertex operator algebra with respect to the finite automorphism
group. As we shall see in our paper, some vertex algebras admit both realizations, coset and
orbifold.
In this paper we consider certain coset vertex algebras for vertex algebras associated to affine
Lie algebras. Let g be a simple Lie algebra of type Xn , ĝ the associated affine Lie algebra of
(1)
type Xn , and LX (1) (kΛ0 ) the simple vertex operator algebra associated to ĝ of level k ∈ C,
n
k 6= −h∨ . For k, m ∈ Z>0 , LX (1) ((k + m)Λ0 ) is a subalgebra of LX (1) (kΛ0 ) ⊗ LX (1) (mΛ0 ), and
n
n
n
one has the associated coset vertex operator algebra
Com(LX (1) ((k + m)Λ0 ), LX (1) (kΛ0 ) ⊗ LX (1) (mΛ0 )).
n
n
n
(1)
Although there are no precise general results (to the best of our knowledge) about the structure
of these cosets, it is believed that these vertex operator algebras are finitely generated and
rational.
2
D. Adamović and O. Perše
In [7], we present a vertex-algebraic proof of the fact that in the case k = 1 and affine
(1)
(1)
Lie algebras of types Dn and Bn , the coset (1) is isomorphic to the rational vertex operator
algebra VL+ .
The situation is even more complicated for general k, m ∈ C, such that k, m, k + m 6= −h∨ .
Then one has the coset vertex operator algebra
e (1) ((k + m)Λ0 ), L (1) (kΛ0 ) ⊗ L (1) (mΛ0 ) ,
Com L
X
X
X
n
n
n
e (1) ((k + m)Λ0 ) is a certain affine vertex operator algebra associated to Xn(1) of level
where L
Xn
k + m (not necessarily simple). In this paper we identify some special cases of such cosets, and
it turns out that they are not rational.
The construction in [7] is based on fermionic construction of vertex algebras and certain
conformal embeddings. In the present paper we use bosonic construction of vertex algebras
and construct new conformal embeddings of affine vertex algebras at level −1. By applying
the bosonic realization of the affine vertex algebras LA(1) (− 21 Λ0 ) and LC (1) (− 12 Λ0 ) (cf. [17]) we
1
`
consider coset vertex algebras
Com LA(1) (−Λ0 ), LA(1) (− 21 Λ0 ) ⊗ LA(1) (− 12 Λ0 )
1
1
1
e (1) (−Λ0 ), L (1) (− 1 Λ0 ) ⊗ L (1) (− 1 Λ0 ) .
Com L
2
2
C
C
C
`
`
and
(2)
`
It is interesting that these cosets have central charge 1. We show that these cosets are isomorphic
to M (1)+ , where M (1) is the Heisenberg vertex operator algebra of rank 1, and M (1)+ is the
Z2 -orbifold vertex algebra studied in [15]. The structure theory of M (1)+ shows that these
cosets are irrational vertex operator algebras and isomorphic to W (2, 4)-algebra with central
charge c = 1.
By combining results from [1] and the present paper, we classify irreducible ordinary
e (1) (−Λ0 )-modules is related
e
LC (1) (−Λ0 )-modules. We believe that the (tensor) category of L
C
`
`
to the (tensor) category of M (1)+ -modules. We plan to address this correspondence in our
forthcoming publications.
Generalizing (2), we use a natural realization of the vertex operator algebra LA(1) (−Λ0 )⊗` as
1
a subalgebra of LC (1) (− 12 Λ0 ) ⊗ LC (1) (− 12 Λ0 ), and prove that
`
`
Com LA(1) (−Λ0 )⊗` , LC (1) (− 21 Λ0 ) ⊗ LC (1) (− 21 Λ0 )
1
`
`
M (1)+ ,
is isomorphic to
where M (1) is the Heisenberg vertex operator algebra of rank `.
Our construction is based on a new, interesting conformal embedding of affine vertex operator algebras at level −1 which can be of independent interest. We show that LC (1) (−Λ0 ) is
`
conformally embedded into LA(1) (−Λ0 ) and that
2`−1
LA(1) (−Λ0 ) ∼
= LC (1) (−Λ0 ) ⊕ LC (1) (−2Λ0 + Λ2 ),
2`−1
`
`
which implies that LC (1) (−Λ0 ) is a Z2 -orbifold of LA(1) (−Λ0 ).
`
2`−1
(1)
By using conformal embeddings we study certain categories of A2`−1 -modules of level −1
(1)
(1)
from [8] as C` -modules. It turns out that irreducible highest weight A2`−1 -modules
LA(1) (−(n + 1)Λ0 + nΛ1 ) and LA(1) (−(n + 1)Λ0 + nΛ2`−1 ) (n ∈ Z>0 ) are also irreducible
2`−1
2`−1
(1)
as C` -modules. This result is an affine analogue of the isomorphism of finite-dimensional
C` -modules:
VA2`−1 (nω2`−1 ) ∼
= VA2`−1 (nω1 ) ∼
= VC` (nω1 ).
The Vertex Algebra M (1)+ and Certain Affine Vertex Algebras of Level −1
3
Using these conformal embeddings we also show that the coset
Com LA(1) (−Λ0 )⊗` , LC (1) (−Λ0 )
1
`
is isomorphic to M (1)+ , where M (1) is the Heisenberg vertex operator algebra of rank ` − 1.
2
Preliminaries
Let V be a vertex algebra [10, 19, 20, 27]. For a subalgebra U of V denote by
Com(U, V ) = {v ∈ V | un v = 0 for all u ∈ U, n ≥ 0}
the commutant of U in V (cf. [21, 22, 27]). Then, Com(U, V ) is a subalgebra of V (also called
coset vertex algebra).
Let h be a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear
form h·, ·i, considered as an Abelian Lie algebra. Let ĥ = h ⊗ C[t, t−1 ] ⊕ CK be its affinization
with the center K. Then the free bosonic Fock space M (1) = S(h ⊗ t−1 C[t−1 ]) is a simple vertex
operator algebra of central charge ` = dim h, with Virasoro vector
`
1 X (i)
ω=
h (−1)2 1,
2
i=1
where {h(1) , . . . , h(`) } is any orthonormal basis of h (cf. [20, 27]). We shall also use the notation
Mh (1) to emphasize the associated vector space h.
Vertex algebra M (1) has an order 2 automorphism which is lifted from the map h 7→ −h, for
h ∈ h. Denote by M (1)+ (or Mh (1)+ ) the subalgebra of invariants of that automorphism. The
irreducible modules for M (1)+ were classified in [15] and [16]. For ` = 1, it was proved in [12]
that M (1)+ is generated by ω and one primary vector of conformal weight 4, so it is isomorphic
to a W (2, 4)-algebra with central charge 1.
Let g be the simple Lie algebra of type Xn , and ĝ the associated affine Lie algebra of type
(1)
Xn . For any weight Λ of ĝ, denote by LX (1) (Λ) the irreducible highest weight ĝ-module. Denote
n
by Λi , i = 0, . . . , n the fundamental weights of ĝ (cf. [23]). We shall also use the notation VXn (µ)
for a highest weight g-module of highest weight µ, and ωi , i = 1, . . . , n for the fundamental
weights of g.
For any k ∈ C, denote by NX (1) (kΛ0 ) the generalized Verma ĝ-module with highest
n
weight kΛ0 . Then, NX (1) (kΛ0 ) is a vertex operator algebra of central charge
n
k 6= −h∨ , with Virasoro vector obtained by Sugawara construction:
dim
Xg
1
ω=
ai (−1)bi (−1)1,
2(k + h∨ )
k dim g
k+h∨ ,
for any
(3)
i=1
where {ai }i=1,...,dim g is an arbitrary basis of g, and {bi }i=1,...,dim g the corresponding dual basis
of g with respect to the symmetric
invariant bilinear form, normalized by the condition that the
√
length of the highest root is 2 (cf. [18, 21, 24, 27, 28]).
It follows that any quotient of NX (1) (kΛ0 ) is a vertex operator algebra, for k 6= −h∨ . Specially,
n
LX (1) (kΛ0 ) is a simple vertex operator algebra, for any k 6= −h∨ .
n
4
D. Adamović and O. Perše
3
Simple Lie algebras of type C` and A2`−1
+
−
2`
Consider two 2`-dimensional vector spaces A1 = ⊕2`
i=1 Cai , A2 = ⊕i=1 Cai and let A = A1 ⊕A2 .
The Weyl algebra W2` is the complex associative algebra generated by A with non-trivial relations
−
[a+
i , aj ] = δi,j ,
1 ≤ i, j ≤ 2`.
The normal ordering on A is defined by
1
: xy := (xy + yx),
2
x, y ∈ A.
Define
+ −
eA
i −j =: ai aj :,
−
fAi −j =: a+
j ai :,
−
Hi = − : a+
i ai :,
1 ≤ i ≤ 2`.
1 ≤ i, j ≤ 2`,
i < j,
and
Then the Lie algebra g1 generated by the set
A
ei −j , fAi −j | 1 ≤ i, j ≤ 2`, i < j
is the simple Lie algebra of type A2`−1 (cf. [11] and [17]). The Cartan subalgebra h1 is spanned
by
{Hi − Hi+1 | 1 ≤ i ≤ 2` − 1}.
Let θ be the automorphism of W2` of order two given by
−
a+
i 7→ a2`+1−i ,
+
a−
2`+1−i 7→ ai ,
+
a−
i 7→ −a2`+1−i ,
−
a+
2`+1−i 7→ −ai ,
for i = 1, . . . , `. Clearly, g1 is θ-stable and
A
θ(eA
i −j ) = −e2`+1−j −2`+1−i ,
A
θ(eA
i −2`+1−j ) = ej −2`+1−i ,
and similarly for root vectors associated to negative roots.
The subalgebra g of g1 generated by
−
+
e2i = a+
f2i = a−
i a2`+1−i ,
i a2`+1−i ,
1
1
−
+
a−
+ a+
fi +j = (a−
a+
+ a−
ei +j = (a+
j a2`+1−i ),
j a2`+1−i ),
2 i 2`+1−j
2 i 2`+1−j
1
1
−
+
−
−
ei −j = (a+
fi −j = (a+
a− − a+
i aj − a2`+1−j a2`+1−i ),
2`+1−i a2`+1−j ),
2
2 j i
for i, j = 1, . . . , `, i < j, is the simple Lie algebra of type C` . The Cartan subalgebra h is
spanned by
{Hi − H2`+1−i | 1 ≤ i ≤ `}.
Clearly, θ acts as 1 on g. Furthermore,
1
−
e∗1 +2 = (a+
a− − a+
2 a2` ) ∈ g1
2 1 2`−1
The Vertex Algebra M (1)+ and Certain Affine Vertex Algebras of Level −1
5
is a highest weight vector for g, which generates the irreducible g-module VC` (ω2 ). Clearly,
θ acts as −1 on VC` (ω2 ). We obtain the decomposition
g1 ∼
= g ⊕ VC` (ω2 ).
(4)
Since
dim VA2`−1 (nω1 ) = dim VC` (nω1 ),
one easily concludes that the irreducible g1 -module VA2`−1 (nω1 ) remains irreducible when restricted to g. Thus,
VA2`−1 (nω1 ) ∼
= VC` (nω1 )
for n ∈ Z≥0 .
(5)
Similarly
VA2`−1 (nω2`−1 ) ∼
= VC` (nω1 )
for n ∈ Z≥0 .
(6)
We will consider certain affine analogues of relations (4)–(6).
In what follows we shall need the following decompositions of g-modules:
VC` (ω2 ) ⊗ VC` (ω2 ) ∼
= VC` (2ω2 ) ⊕ VC` (ω1 + ω3 ) ⊕ VC` (ω4 )
⊕ VC` (2ω1 ) ⊕ VC` (ω2 ) ⊕ VC` (0)
(` ≥ 4),
VC3 (ω2 ) ⊗ VC3 (ω2 ) ∼
= VC3 (2ω2 ) ⊕ VC3 (ω1 + ω3 ) ⊕ VC3 (2ω1 ) ⊕ VC3 (ω2 ) ⊕ VC3 (0),
VC (ω2 ) ⊗ VC (ω2 ) ∼
= VC (2ω2 ) ⊕ VC (2ω1 ) ⊕ VC (0).
2
4
2
2
2
2
(7)
Weyl vertex algebras and symplectic af f ine Lie algebras
The Weyl algebra W` ( 12 + Z) is a complex associative algebra generated by
a±
i (r),
r∈
1
2
+ Z,
1≤i≤`
with non-trivial relations
−
[a+
i (r), aj (s)] = δr+s,0 δi,j ,
where r, s ∈ 12 + Z, i, j ∈ {1, . . . , `}.
Let M` be the irreducible W` ( 12 + Z)-module generated by the cyclic vector 1 such that
a±
i (r)1 = 0
for r > 0,
1 ≤ i ≤ `.
Define the following fields on M`
X
1 −n−1
a±
(z)
=
a±
.
i
i (n + 2 )z
n∈Z
The fields a±
i (z), i = 1, . . . , ` generate on M` the unique structure of a simple vertex algebra
(cf. [18, 24]). Let us denote the corresponding vertex operator by Y .
We have the following Virasoro vector in M` :
ω=
`
1X − 3 + 1
3 −
1
ai (− 2 )ai (− 2 ) − a+
i (− 2 )ai (− 2 ) 1.
2
i=1
(8)
6
D. Adamović and O. Perše
Let Y (ω, z) =
P
n∈Z
L(n)z −n−2 . Then M` is 12 Z≥0 -graded with respect to L(0):
M
M` :=
M` (m) = {v ∈ M` | L(0)v = mv}.
M` (m),
1
m∈ 2 Z≥0
Note that M` (0) = C1. For v ∈ M` (m) we shall write wt(v) = m.
The following result is well-known.
Theorem 1 ([17]). We have
M
M` ∼
LC (1) (− 23 Λ0 + Λ1 ).
= LC (1) (− 12 Λ0 )
`
`
In the case ` = 1 we have
M
LA(1) (− 32 Λ0 + Λ1 ).
M1 ∼
= LA(1) (− 12 Λ0 )
1
1
Remark 1. The highest weights of modules from Theorem 1 are admissible in the sense of [25].
(1)
(1)
Representations of vertex operator algebras associated to affine Lie algebras of type A1 and C`
with admissible highest weights were studied in [5] and [2].
We shall now consider the vertex algebra M2` and its subalgebra LC (1) (− 21 Λ0 )⊗LC (1) (− 12 Λ0 ).
`
`
For ` = 1, LA(1) (− 12 Λ0 ) ⊗ LA(1) (− 12 Λ0 ) is a subalgebra of M2 .
1
1
Let θ : M2` → M2` be the automorphism of order two of the vertex algebra M2` which is
lifted from the following automorphism of the Weyl algebra
−
a+
i (s) 7→ a2`+1−i (s),
+
a−
2`+1−i (s) 7→ ai (s),
+
a−
i (s) 7→ −a2`+1−i (s),
−
a+
2`+1−i (s) 7→ −ai (s),
for i = 1, . . . , ` and s ∈ 21 + Z.
If we have a subalgebra U ⊂ M2` which is θ-stable, we define
U 0 = {u ∈ U | θ(u) = u},
U 1 = {u ∈ U | θ(u) = −u}.
Define the following vectors in M2` :
1
−
1
1
= √ (a+
i (− 2 ) + a2`+1−i (− 2 ))1,
2
1
−
+
1
1
b−
i = √ (ai (− 2 ) − a2`+1−i (− 2 ))1,
2
b+
i
b+
2`+1−i
b−
2`+1−i
√
−1
−
1
1
= √ (a+
i (− 2 ) − a2`+1−i (− 2 ))1,
2
√
− −1 − 1
1
= √ (ai (− 2 ) + a+
2`+1−i (− 2 ))1,
2
−
+
−
for i = 1, . . . , `. Then the subalgebra generated by b+
i , bi (resp. b2`+1−i , b2`+1−i ), for i = 1, . . . , `,
is isomorphic to M` . Since
±
θ(b±
i ) = bi ,
±
θ(b±
2`+1−i ) = −b2`+1−i ,
for i = 1, . . . , `, we have
0 ∼
M2`
= M` ⊗ M`0 ∼
= M` ⊗ LC (1) (− 12 Λ0 ),
`
and for ` = 1:
M20 ∼
= M1 ⊗ M10 ∼
= M1 ⊗ LA(1) (− 12 Λ0 ).
1
The Vertex Algebra M (1)+ and Certain Affine Vertex Algebras of Level −1
7
Commutant of LA(1) (−Λ0 )⊗` in LC (1) (− 12 Λ0 ) ⊗ LC (1) (− 12 Λ0 )
5
1
`
`
In this section we use Weyl vertex algebra M2` to study certain subalgebras of LC (1) (− 21 Λ0 ) ⊗
`
LC (1) (− 12 Λ0 ). We determine the commutants
`
Com LA(1) (−Λ0 )⊗` , LC (1) (− 12 Λ0 ) ⊗ LC (1) (− 21 Λ0 )
1
`
`
and
Com LA(1) (−Λ0 ), LA(1) (− 21 Λ0 ) ⊗ LA(1) (− 12 Λ0 ) .
1
1
1
0 generated by
Let U be the vertex subalgebra of M`0 ⊗ M`0 ⊂ M2`
1
1 −
e(i) = a+
i (− 2 )a2`+1−i (− 2 )1,
1
1 +
f (i) = a−
i (− 2 )a2`+1−i (− 2 )1
and
+
1 −
1
1
1 −
h(i) = (−a+
i (− 2 )ai (− 2 ) + a2`+1−i (− 2 )a2`+1−i (− 2 ))1,
(9)
i = 1, . . . , `. It is clear that U is isomorphic to the tensor product
LA(1) (−Λ0 ) ⊗ · · · ⊗ LA(1) (−Λ0 )
1
{z
}
| 1
` times
of ` copies of the affine vertex algebra LA(1) (−Λ0 ).
1
Define also
+
1 −
1 −
1
1
H (i) = (a+
i (− 2 )ai (− 2 ) + a2`+1−i (− 2 )a2`+1−i (− 2 ))1,
i = 1, . . . , `.
Then
H (i) ∈ Com(U, M2` ),
i = 1, . . . , `.
Let
h=
`
M
CH (i) .
i=1
Then h can be considered as an Abelian Lie algebra, and the components of the vertex operators
X
Y (h, z) =
h(n)z −n−1 ,
h ∈ h,
n∈Z
define a representation of the associated Heisenberg algebra b
h. Moreover, h generates the subalgebra of the M2` which is isomorphic to the Heisenberg vertex algebra Mh (1) with central
charge `. We also note that hH (i) , H (j) i = H (i) (1)H (j) = −2δij .
Using relations (9) one can show that the Virasoro vector (8) (in M2` ) can be written in
a form:
ω = ω1 + ω2
where
`
1 X (i)
e (−1)f (i) (−1) + f (i) (−1)e(i) (−1) + 12 h(i) (−1)2 1
ω1 =
2
i=1
8
D. Adamović and O. Perše
is the Virasoro vector in U and
`
1 X (i)
ω2 = −
H (−1)2 1
4
i=1
is the Virasoro vector in Mh (1). For i = 1, 2 let Y (ωi , z) =
Li (n)z −n−2 .
P
n∈Z
Proposition 1. We have:
Com(U, M2` ) = Mh (1).
Proof . It is clear that W = Com(U, M2` ) contains a subalgebra isomorphic to Mh (1).
Assume now that Mh (1) 6= W . Then there is a vector w ∈ W , wt(w) > 0 such that
H (i) (n)w = δn,0 λ(i) w,
n ∈ Z≥0 ,
λ(i) ∈ Z,
i = 1, . . . , `.
(Note that each H (i) (0) acts semisimply on M2` with eigenvalues in Z.) By the definition of W
we have that L1 (0)w = 0. Therefore
`
1X
(λ(i) )2 w.
L(0)w = L2 (0)w = −
4
i=1
This contradicts the fact that wt(w) > 0. So, W = Mh (1).
Since θ(H (i) ) = −H (i) for i = 1, . . . , ` we have that Mh (1)+ = Mh (1)0 is a subalgebra of
⊗ M`0 . Therefore the vertex algebra LC (1) (− 21 Λ0 ) ⊗ LC (1) (− 12 Λ0 ) contains a subalgebra
M`0
`
`
isomorphic to U ⊗ Mh (1)+ . By using Proposition 1 we obtain the following theorem.
Theorem 2. We have:
Com LA(1) (−Λ0 )⊗` , LC (1) (− 21 Λ0 ) ⊗ LC (1) (− 21 Λ0 ) ∼
= Mh (1)+ .
1
`
`
In the case ` = 1, h is a one-dimensional vector space h = CH, where
+
1
1 −
1
1 −
H = a+
1 (− 2 )a1 (− 2 ) + a2 (− 2 )a2 (− 2 ) 1.
Theorem 3. We have:
Com LA(1) (−Λ0 ), LA(1) (− 21 Λ0 ) ⊗ LA(1) (− 12 Λ0 ) ∼
= Mh (1)+ .
1
1
1
In the next section we generalize Theorem 3 in another way.
6
(1)
Commutant of level −1 type C`
in LC (1) (− 21 Λ0 ) ⊗ LC (1) (− 12 Λ0 )
`
af f ine vertex algebra
`
In this section we study another subalgebra of LC (1) (− 12 Λ0 ) ⊗ LC (1) (− 21 Λ0 ). The subalgebra of
0 generated by
M`0 ⊗ M`0 ⊂ M2`
1 −
1
e2i (= e(i) ) = a+
i (− 2 )a2`+1−i (− 2 )1,
ei +j =
fi +j =
1
2
1
2
`
`
1 +
1
f2i (= f (i) ) = a−
i (− 2 )a2`+1−i (− 2 )1,
+
1 −
1
1 −
1
a+
i (− 2 )a2`+1−j (− 2 ) + aj (− 2 )a2`+1−i (− 2 ) 1,
−
1 +
1
1 +
1
a−
i (− 2 )a2`+1−j (− 2 ) + aj (− 2 )a2`+1−i (− 2 ) 1,
The Vertex Algebra M (1)+ and Certain Affine Vertex Algebras of Level −1
ei −j =
fi −j =
1
2
1
2
9
+
1 −
1
1 −
1
a+
)a
)
−
a
(−
)a
(−
)
(−
(−
i
2`+1−j
2 j
2
2 2`+1−i
2 1,
+
+
1 −
1 −
1
aj (− 2 )ai (− 2 ) − a2`+1−i (− 2 )a2`+1−j (− 21 ) 1,
for i, j = 1, . . . , `, i < j,
(10)
(1)
e (1) (−Λ0 ).
is a level −1 affine vertex operator algebra associated to C` . We denote it by L
C`
Let
H=
`
X
H (i) =
i=1
2`
X
1
1 −
a+
i (− 2 )ai (− 2 )1.
i=1
Set h1 = CH ⊂ h. Let Mh1 (1) be the Heisenberg vertex algebra generated by H. Clearly
hH, Hi = −2`.
e (1) (−Λ0 ), M2` ), we have that M2` contains a subalgebra isomorphic to
Since H ∈ Com(L
C
`
e (1) (−Λ0 ) ⊗ Mh (1).
L
1
C
`
Proposition 2. The Virasoro vector (8) in M2` can be written in a form:
ω = ω1 + ω2 ,
e (1) (−Λ0 ) obtained by the Sugawara construction and
where ω1 is the Virasoro vector (3) in L
C
`
ω2 = −
1
H(−1)2 1
4`
is the Virasoro vector in Mh1 (1).
Proof . Formula (3) implies that
ω1 =
1
2`
+2
`
X
(e2i (−1)f2i (−1) + f2i (−1)e2i (−1))1
i=1
`
X
(ei +j (−1)fi +j (−1) + fi +j (−1)ei +j (−1))1
i,j=1
i<j
`
X
!
`
1X
h2i (−1)2 1 ,
+2
(ei −j (−1)fi −j (−1) + fi −j (−1)ei −j (−1))1 +
2
i,j=1
i<j
(11)
i=1
where
+
1 −
1
1 −
1
h2i (= h(i) ) = −a+
i (− 2 )ai (− 2 ) + a2`+1−i (− 2 )a2`+1−i (− 2 ) 1.
It follows from relations (10) that
(e2i (−1)f2i (−1) + f2i (−1)e2i (−1))1
1 −
1 +
1 −
1
= 2a+
i (− 2 )ai (− 2 )a2`+1−i (− 2 )a2`+1−i (− 2 )1
−
3 +
1
3 +
1
+ a−
i (− 2 )ai (− 2 )1 + a2`+1−i (− 2 )a2`+1−i (− 2 )1
+
3 −
1
3 −
1
− a+
i (− 2 )ai (− 2 )1 − a2`+1−i (− 2 )a2`+1−i (− 2 )1,
2(ei +j (−1)fi +j (−1) + fi +j (−1)ei +j (−1))1
1 −
1 +
1 −
1
= a+
i (− 2 )ai (− 2 )a2`+1−j (− 2 )a2`+1−j (− 2 )1
(12)
10
D. Adamović and O. Perše
1 −
1 +
1
1 −
+ a+
i (− 2 )aj (− 2 )a2`+1−j (− 2 )a2`+1−i (− 2 )1
1 +
1
1 −
1 +
+ a−
i (− 2 )aj (− 2 )a2`+1−i (− 2 )a2`+1−j (− 2 )1
1 −
1 −
1
1 +
+ a+
j (− 2 )aj (− 2 )a2`+1−i (− 2 )a2`+1−i (− 2 )1
1 − 3 + 1
−
3 +
1
3 +
1
a (− 2 )ai (− 2 )1 + a−
+
j (− 2 )aj (− 2 )1 + a2`+1−i (− 2 )a2`+1−i (− 2 )1
2 i
+
+
1
3 +
3 −
1
3 −
1
+ a−
2`+1−j (− 2 )a2`+1−j (− 2 )1 − ai (− 2 )ai (− 2 )1 − aj (− 2 )aj (− 2 )1
+
3 −
1
3 −
1
− a+
2`+1−i (− 2 )a2`+1−i (− 2 )1 − a2`+1−j (− 2 )a2`+1−j (− 2 )1 ,
(13)
and
2(ei −j (−1)fi −j (−1) + fi −j (−1)ei −j (−1))1
1 −
1 +
1 −
1
= a+
i (− 2 )ai (− 2 )aj (− 2 )aj (− 2 )1
1 −
1
1 −
1 +
− a+
j (− 2 )ai (− 2 )a2`+1−j (− 2 )a2`+1−i (− 2 )1
1 −
1
1 +
1 −
− a+
i (− 2 )aj (− 2 )a2`+1−i (− 2 )a2`+1−j (− 2 )1
1 −
1 +
1 −
1
+ a+
2`+1−i (− 2 )a2`+1−i (− 2 )a2`+1−j (− 2 )a2`+1−j (− 2 )1
1 − 3 + 1
−
3 +
1
3 +
1
+
a (− 2 )ai (− 2 )1 + a−
j (− 2 )aj (− 2 )1 + a2`+1−i (− 2 )a2`+1−i (− 2 )1
2 i
+
+
3 +
1
3 −
1
3 −
1
+ a−
2`+1−j (− 2 )a2`+1−j (− 2 )1 − ai (− 2 )ai (− 2 )1 − aj (− 2 )aj (− 2 )1
+
3 −
1
3 −
1
− a+
(−
)a
(−
)1
−
a
(−
)a
(−
)1
,
2`+1−i
2`+1−j
2 2`+1−i
2
2 2`+1−j
2
(14)
for all i, j = 1, . . . , `, i < j. Using relations (11), (12), (13) and (14) one can obtain
ω1 =
2`
1X − 3 + 1
1
3 −
1
ai (− 2 )ai (− 2 ) − a+
H(−1)2 1,
i (− 2 )ai (− 2 ) 1 +
2
4`
i=1
which implies the claim of proposition.
Using Proposition 2 and applying similar arguments as in the proof of Proposition 1, we
obtain:
Proposition 3. We have:
e (1) (−Λ0 ), M2` = Mh (1).
Com L
1
C
`
Since θ(H) = −H, we have that Mh1 (1)+ = Mh1 (1)0 is a subalgebra of M`0 ⊗ M`0 . Therefore
e (1) (−Λ0 )⊗
the vertex algebra LC (1) (− 21 Λ0 )⊗LC (1) (− 21 Λ0 ) contains a subalgebra isomorphic to L
C
`
`
`
Mh1 (1)+ . By using Proposition 3 we obtain the following theorem.
Theorem 4. We have:
∼ Mh (1)+ .
e (1) (−Λ0 ), L (1) (− 1 Λ0 ) ⊗ L (1) (− 1 Λ0 ) =
Com L
1
2
2
C
C
C
`
7
`
`
e (1) (−Λ0 )
The classif ication of ordinary modules for L
C
`
e (1) (−Λ0 )-modules, which we use in the
In this section we obtain a classification of irreducible L
C
following sections.
`
The Vertex Algebra M (1)+ and Certain Affine Vertex Algebras of Level −1
11
The following vertex algebra was considered in [1]: Let ` ≥ 3 and
VC (1) (−Λ0 ) =
`
NC (1) (−Λ0 )
`
h∆3 (−1)1i
,
where h∆3 (−1)1i is the ideal generated by the singular vector ∆3 (−1)1, such that ∆3 (−1) is
given by the following determinant:
e21 (−1) e1 +2 (−1) e1 +3 (−1) ∆3 (−1) = e1 +2 (−1) e22 (−1) e2 +3 (−1) .
e + (−1) e + (−1) e2 (−1) 3
2
3
1
3
Using relations (10), one can easily check that
e21 (−1) e1 +2 (−1) e1 +3 (−1) e1 +2 (−1) e22 (−1) e2 +3 (−1) 1 = 0
e + (−1) e + (−1) e2 (−1) 3
2
3
1
3
e (1) (−Λ0 ), so L
e (1) (−Λ0 ) is a certain quotient of V (1) (−Λ0 ), for ` ≥ 3. Thus, any irrein L
C
C
C
`
`
`
e (1) (−Λ0 ) is an irreducible module for V (1) (−Λ0 ). The classification of
ducible module for L
C
C
`
`
all irreducible modules in the category O for VC (1) (−Λ0 ) was obtained in [1, Example 4.1]. We
3
e (1) (−Λ0 )-modules.
will apply this result to obtain a classification of all irreducible ordinary L
C
`
(Recall that a module is called ordinary if L(0) acts semisimply with finite-dimensional weight
spaces.)
Proposition 4. Let ` ≥ 3. The set
{LC (1) ((−n − 1)Λ0 + nΛ1 ) | n ∈ Z≥0 } ∪ {LC (1) (−2Λ0 + Λ2 )}
(15)
`
`
provides a complete list of irreducible ordinary modules for the vertex operator algebras
e (1) (−Λ0 ).
VC (1) (−Λ0 ) and L
C
`
`
Proof . We use the well-known method for classification of highest weights of VC (1) (−Λ0 )`
modules as solutions of certain polynomial equations arising from the singular vectors (cf. [2,
5, 8, 29, 30, 31]). The highest weights of ordinary modules are of the form −Λ0 + µ, where
P̀
µ=
hi i . Clearly, hi ∈ Z≥0 , for i = 1, . . . , `. Using polynomials from [1, Example 4.1], and
i=1
the fact that
e21 (−1) e1 +2 (−1) e1 +i (−1)
2
1
e1 +2 (−1) e22 (−1) e2 + (−1)
f
(0)
∆
(−1)1
=
−
3
3
i
i
2
e + (−1) e + (−1) e2 (−1)
1
2
i
i
i
1,
for i = 4, . . . , `, we obtain that the weights µ are annihilated by the polynomials
pi (µ) = (h1 + 1)(h2 + 21 )hi ,
qi (µ) = (h1 + 1)(4hi + (h2 + hi )(h2 + hi − 1)),
ri (µ) = 4hi (h2 + 1) + (h1 + hi − 1)(h2 + hi + h2 (h1 + hi )),
for i = 3, . . . , `. Then one easily obtains that hi = 0, for i = 3, . . . , `, and that either h2 = 0
and h1 = n, for n ∈ Z≥0 or h2 = 1 and h1 = 1. So we have proved that any irreducible ordinary
e (1) (−Λ0 )-module) must belong to the set (15).
VC (1) (−Λ0 )-module (resp. L
C
`
`
12
D. Adamović and O. Perše
n
Since a+
1 (−1/2) 1 ∈ M2` is a singular vector of highest weight −(n + 1)Λ0 + nΛ1 , and
+
1 −
1
1 −
1
e∗1 +2 = 21 (a+
1 (− 2 )a2`−1 (− 2 )1 − a2 (− 2 )a2` (− 2 )1) ∈ M2`
is a singular vector of highest weight −2Λ0 + Λ2 , we have that every module from the set (15)
is a module for these vertex operator algebras. The proof is now complete.
e (1) (−Λ0 ) is simple.
Remark 2. In our new paper [6] we prove that the vertex operator algebra L
C
`
Therefore, Proposition 4 also gives the classification of irreducible LC (1) (−Λ0 )-modules.
`
8
(1)
Conformal embedding of C`
(1)
into A2`−1 at level −1
In this section we show that LC (1) (−Λ0 ) is a Z2 -orbifold of vertex operator algebra LA(1) (−Λ0 ),
`
2`−1
and determine the corresponding decomposition.
The subalgebra of M2` generated by
+
1
1 −
eA
i −j = ai (− 2 )aj (− 2 )1,
1 +
1
fAi −j = a−
i (− 2 )aj (− 2 )1
for i, j = 1, . . . , 2`,
i < j,
(1)
is a level −1 affine vertex operator algebra associated to the affine Lie algebra ĝ1 of type A2`−1 .
e (1) (−Λ0 ). Clearly, L
e (1) (−Λ0 ) is a subalgebra of L
e (1) (−Λ0 ). Let ĝ be the
We denote it by L
A
C
A
2`−1
`
2`−1
(1)
affine Lie algebra of type C` .
e (1) (−Λ0 ), and by ω A the Virasoro vector in
As before, denote by ω1 the Virasoro vector in L
1
C
`
e (1) (−Λ0 ).
L
A
2`−1
Proposition 5. We have
ω1 = ω1A .
Proof . Similarly as in Proposition 2, one can show that
ω1A
2`
1X − 3 + 1
1
3 −
1
=
ai (− 2 )ai (− 2 ) − a+
H(−1)2 1,
i (− 2 )ai (− 2 ) 1 +
2
4`
i=1
which implies the claim of proposition.
Furthermore, the vector e∗1 +2 from the proof of Proposition 4 is a singular vector for ĝ in
e (1) (−Λ0 ) which generates L
e (1) (−Λ0 )-module L
e (1) (−2Λ0 + Λ2 ), whose top component is
L
A
C
C
2`−1
`
`
e (1) (−Λ0 ) and as −1 on L
e (1) (−2Λ0 +Λ2 ).
irreducible g-module VC` (ω2 ). Clearly, θ acts as 1 on L
C
C
`
`
e (1) (−2Λ0 + Λ2 ). Then un v ∈ L
e (1) (−Λ0 ), for any n ∈ Z.
Lemma 1. Let u, v ∈ L
C
C
`
`
e (1) (−2Λ0 +Λ2 ).
Proof . It suffices to prove the lemma for u and v from top component R(0) of L
C
Then the statement will follow from the associator formulae.
First we notice that
`
e (1) (−Λ0 ),
u0 v ∈ L
C
`
(since vectors of conformal weight 1 with bracket [u, v] = u0 v span Lie algebra g1 of type A2`−1 ,
and fixed point subalgebra g is a Lie algebra of type C` ).
The Vertex Algebra M (1)+ and Certain Affine Vertex Algebras of Level −1
13
Assume now that
e (1) (−Λ0 )
/L
un0 v ∈
C
`
for certain u, v ∈ R(0) and n0 ∈ Z. Take maximal n0 with this property.
e (1) (−Λ0 )-module W of highest
Then un0 v has nontrivial component in some highest weight L
C
`
weight −Λ0 + µ, and therefore there is a nontrivial intertwining operator of type
W
e (1) (−2Λ0 + Λ2 ) L
e (1) (−2Λ0 + Λ2 ) .
L
C`
C`
One can associate to this intertwining operator, a non-trivial g-homomorphism
f : VC` (ω2 ) ⊗ VC` (ω2 ) → VC` (µ).
In particular, VC` (µ) must appear in the decomposition of tensor product VC` (ω2 ) ⊗ VC` (ω2 ).
First consider the case ` = 2. Using the decomposition of tensor product VC2 (ω2 ) ⊗ VC2 (ω2 )
from (7) and the fact that the lowest conformal weights of modules of highest weights −Λ0 + 2ω2
and −Λ0 + 2ω1 are 25 and 23 , respectively, we conclude that these modules cannot appear inside
e (1) (−Λ0 ). Thus, un v ∈ L
e (1) (−Λ0 ).
L
0
A
C
2`−1
`
Now, let ` ≥ 3. Relation (7) implies that the only cases of weights −Λ0 + µ (aside from
µ = 0) from Proposition 4 such that µ appears in the decomposition of VC` (ω2 ) ⊗ VC` (ω2 ) are
when µ = 2ω1 or µ = ω2 . Since un0 v is θ-invariant, it does not contain component inside
e (1) (−2Λ0 + Λ2 ). On the other hand, the lowest conformal weight of module of highest weight
L
C
`
−Λ0 + 2ω1 is
`+1
` ,
e (1) (−Λ0 ).
which is not an integer. Thus, this module cannot appear inside L
A
2`−1
e (1) (−Λ0 ).
Thus, un0 v ∈ L
C
`
Theorem 5. We have:
e (1) (−Λ0 ) = L
e (1) (−Λ0 ) ⊕ L
e (1) (−2Λ0 + Λ2 ).
L
A
C
C
2`−1
`
`
In particular,
e (1) (−Λ0 ) = L
e (1) (−Λ0 )0 ,
L
C
A
`
e (1) (−2Λ0 + Λ2 ) = L
e (1) (−Λ0 )1 .
L
C
A
2`−1
`
2`−1
e (1) (−2Λ0 + Λ2 ) is a vertex subalgebra of
e (1) (−Λ0 ) ⊕ L
Proof . Lemma 1 shows that L
C
C
`
`
e (1) (−Λ0 ). But this subalgebra clearly contains generators of L
e (1) (−Λ0 ), which implies
L
A
A
2`−1
2`−1
the claim of theorem.
e (1) (−Λ0 )-modules follows from the results
The classification of irreducible ordinary L
A
2`−1
from [8] and similar arguments as in Section 7:
Proposition 6. The set
LA(1) ((−n − 1)Λ0 + nΛ1 ) | n ∈ Z≥0 ∪ LA(1) ((−n − 1)Λ0 + nΛ2`−1 ) | n ∈ Z≥0
2`−1
2`−1
provides a complete list of irreducible ordinary modules for the vertex operator algebra
e (1) (−Λ0 ).
L
A
2`−1
14
D. Adamović and O. Perše
e (1) (−Λ0 )-modules remain irreducible
The following result shows that most irreducible L
A
2`−1
e (1) (−Λ0 ).
when we restrict them on L
C
`
Proposition 7. Assume that ` ≥ 3, n ∈ Z>0 . Then we have the following isomorphisms of
e (1) (−Λ0 )-modules:
L
C
`
LA(1) (−(n + 1)Λ0 + nΛ1 ) ∼
= LC (1) (−(n + 1)Λ0 + nΛ1 ),
2`−1
LA(1)
2`−1
`
(−(n + 1)Λ0 + nΛ2`−1 ) ∼
= LC (1) (−(n + 1)Λ0 + nΛ1 ).
`
Proof . We use the Theorem 6.1 from [14]. The definition of automorphism θ then implies (in
the notation of [14]) that
θ ◦ LA(1) (−(n + 1)Λ0 + nΛ1 ) ∼
= LA(1) (−(n + 1)Λ0 + nΛ2`−1 ),
2`−1
2`−1
e (1) (−Λ0 )-modules.
as L
A
2`−1
Theorem 6.1 from [14] now implies that LA(1) (−(n + 1)Λ0 + nΛ1 ) and LA(1) (−(n + 1)Λ0 +
2`−1
2`−1
e (1) (−Λ0 )-modules. The claim of Proposition now follows easily. nΛ2`−1 ) are irreducible as L
C
`
e (1) (−Λ0 ) and L
e (1) (−Λ0 ) are simple. We don’t use this result in
In [6] we shall prove that L
A
C
2`−1
`
the present paper. But even without these simplicity results we can conclude that the analogous
of Theorem 5 also holds for simple vertex operator algebras LA(1) (−Λ0 ) and LC (1) (−Λ0 ).
2`−1
`
Corollary 1. We have:
LA(1) (−Λ0 ) = LC (1) (−Λ0 ) ⊕ LC (1) (−2Λ0 + Λ2 ).
2`−1
`
`
Proof . First we notice that the automorphism θ also naturally acts on a simple vertex operator
algebra LA(1) (−Λ0 ). Theorem 5 implies that
2`−1
LA(1) (−Λ0 ) = V 0 ⊕ V 1
2`−1
e (1) (−Λ0 ) (resp. L
e (1) (−2Λ0 + Λ2 )). By using the fact
where V 0 (resp. V 1 ) is a quotient of L
C
C
`
`
that Z2 -orbifold components of simple vertex operator algebra are simple (cf. [14]), we get that
V 0 = LC (1) (−Λ0 ) and V 1 = LC (1) (−2Λ0 + Λ2 ). The proof follows.
`
9
`
Commutant of LA(1) (−Λ0 )⊗` in LC (1) (−Λ0 )
1
`
We shall now study vertex operator algebra LC (1) (−Λ0 ). Corollary 1 implies that LC (1) (−Λ0 )
`
`
is a Z2 -orbifold of vertex operator algebra LA(1) (−Λ0 ). Clearly, LA(1) (−Λ0 ) is a quotient of
2`−1
2`−1
e (1) (−Λ0 ) modulo the maximal submodule of L
e (1) (−Λ0 ) (which is possibly zero).
L
A
A
2`−1
2`−1
As before, we denote
+
1 −
1
1 −
1
H (i) = (a+
i (− 2 )ai (− 2 ) + a2`+1−i (− 2 )a2`+1−i (− 2 ))1,
H = H (1) + · · · + H (`) .
i = 1, . . . , `,
The Vertex Algebra M (1)+ and Certain Affine Vertex Algebras of Level −1
15
Also, let
H
(i)
= H (i) − H (i+1) ,
h1 = CH
(1)
(i)
Clearly hH , H
i = 1, . . . , ` − 1,
+ · · · + CH
(j)
(`−1)
and
.
i = −4δi,j .
Theorem 6. We have:
Com LA(1) (−Λ0 )⊗` , LC (1) (−Λ0 ) ∼
= Mh (1)+ .
1
1
`
Proof . By using same arguments as in the proof of Proposition 1 we get that:
e (1) (−Λ0 ) = M (1).
Com LA(1) (−Λ0 )⊗` , L
h
A
1
1
2`−1
Since generators H
(i)
e (1) (−Λ0 ) (which is
don’t belong to the maximal submodule of L
A
2`−1
possibly zero) we conclude that
∼ M (1).
Com LA(1) (−Λ0 )⊗` , LA(1) (−Λ0 ) =
h1
1
(16)
2`−1
Let θ be the automorphism of order two of the vertex operator algebra LA(1) (−Λ0 ) as in
2`−1
Corollary 1. Then
(i)
(i)
θ(H ) = −H ,
i = 1, . . . , ` − 1,
and we get that
Mh (1)+ ⊂ LA(1) (−Λ0 )0 = LC (1) (−Λ0 ).
1
2`−1
(17)
`
Now proof of the theorem follows from relations (16) and (17).
Acknowledgements
The authors gratefully acknowledge partial support by the Ministry of Science, Education and
Sports of the Republic of Croatia, Project ID 037-0372794-2806.
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