Classification of modules of the intermediate series over the twisted N=2 superconformal algebra

Abstract

In this paper, a classification of modules of the intermediate series over the twisted N=2 superconformal algebra is obtained.

Full text

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arXiv:0812.5001v1 [math.RA] 30 Dec 2008
Classification of modules of the intermediate series
over the twisted N = 2 superconformal algebra1
Junbo Li†,‡), Yucai Su†), Linsheng Zhu‡)
†)DepartmentofMathematics,UniversityofScienceandTechnologyofChina,Hefei230026, China
‡)Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China
E-mail: sd junbo@163.com, ycsu@ustc.edu.cn, lszhu@cslg.edu.cn
Abstract. In this paper, a classification of modules of the intermediate series over the
twisted N = 2 superconformal algebra is obtained.
Key words: N = 2 superconformal algebras, modules of the intermediate series, Harish-
Chandra modules.
MR(2000) Subject Classification: 17B10, 17B65, 17B68.
§1. Introduction
The N = 2 superconformal algebras, constructed independently by Kac [5] and by Ade-
mollo et al. [1], whose structure and representation theories were investigated in a series of
papers (e.g., [2, 3, 6, 8], etc), have attracted more and more attentions in recent years. By
choosing the standard Virasoro generators, one observes that the superconformal algebras
fall into the following four types: (i) the Neveu-Schwarz algebra resulting from antiperiodic
boundary conditions, (ii) the Ramond algebra corresponding to periodic boundary condi-
tions, (iii) the topological N = 2 algebra being as the symmetry algebra of topological
conformal field theory, (iv) the twisted superconformal algebra possessing mixed boundary
conditions. It is well known that the first three superconformal algebras are isomorphic to
each other.
The twisted N = 2 superconformal algebra L consists of the Virasoro algebra generators
Lm, m ∈ Z (corresponding to the stress-energy tensor), the Heisenberg algebra generators
Tr, r ∈
(which are the modes of the two spin-3
1
2+ Z (corresponding to the U(1) current), the fermionic generators Gp, p ∈
2fermionic fields), with the following commutation
relations (where c is a central element; see, e.g., [2])
1
2Z
[Lm,Ln] = (m − n)Ln+m+m3−m
[Lm,Tr] = −rTr+m,
[Lm,Gp] = (m
12δm+n,0c,
[Tr,Ts] =r
3δr+s,0c,
2− p)Gp+m,
?
(−1)2p+1(p − q)Tp+q
[Tr,Gp] = Gp+r,
[Gp,Gq] =
(−1)2p?2Lp+q+1
3(p2−1
4)δp+q,0c?
if p + q ∈ Z,
if p + q ∈1
2+ Z,
(1.1)
for m,n ∈ Z, r,s ∈1
2+ Z, p,q ∈1
2Z.
1Supported by NSF grants 10671027, 10825101 of China, “One Hundred Talents Program” from Uni-
versity of Science and Technology of China.
Corresponding E-mail: sd junbo@163.com
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Obviously, L is Z2-graded: L = L0⊕ L1, with
L0= spanC{Lm, Tr, c|m ∈ Z, r ∈1
2+ Z},
L1= spanC{Gp|p ∈1
2Z}, (1.2)
and the Cartan subalgebra H = CL0+ Cc. One can easily see that L contains the well-
known Virasoro algebra Vir = spanC{Lm,c|m ∈ Z}, the super-Virasoro algebras NS (the
N = 1 Neveu-Schwarz algebra [10]) spanned by {Ln,Gr,c|n ∈ Z,r ∈1
N = 1 Ramond algebra [11]) spanned by {Ln,Gn,c|n ∈ Z}.
An L-module V = V0⊕ V1is called a Harish-Chandra module if Vαis a direct sum of
its finite dimensional weight spaces Vλ
λ ∈ H∗(the dual space of H). Similar to the case of Virasoro algebra, we can define the
module of the intermediate series over L.
Definition 1.1. A Z2-graded module V = V0⊕ V1over L is called a module of the inter-
mediate series if V is a Harish-Chandra module such that dimVλ
2+ Z}, and R (the
α= {v ∈ Vα|xv = λ(x)v, x ∈ H} for α ∈ Z2and all
α? 1 for λ ∈ H∗, α ∈ Z2.
Although much progress on representation theory over the N = 2 superconformal alge-
bras has been made in recent years and many aspects arise that are new to representations of
chiral algebras, there are still many questions (e.g., structure of Verma modules, irreducible
modules, classification of Harish-Chandra modules, etc.). In order to better understand the
representations of the Lie algebra L, it is highly desirable to give a classification of modules
of the intermediate series over L. This is the task of the present paper.
The similar problem has been considered over the Ramond sector in [3]. However, due
to the facts that the twisted sector is not isomorphic to the Ramond sector and the number
of fermionic currents N increases, the modules over the twisted sector are interesting and
complicated. It is well known that the twisted N = 2 algebra behaves very differently from
the other N = 2 algebras with respect to the adapted ordering method and thus gives
more new and different insights in superconformal representation theory. In [2], the authors
created the basis for the study of the embedding structure of the twisted N = 2 highest
weight representations and deduced the singular dimensions as well as the multiplication
rules for singular vector operators, which both are crucial for the analysis of the embedding
structure. In [4], the structures of both Verma modules and Fock modules over the twisted
sector of the N = 2 superconformal algebras were investigated. Because of lack of coset
construction of the twisted sector, the methods used in other sectors are not applicable in
the twisted case.
Now let us formulate the main result below. First we define two families Aa,b, Ba,b, a,b ∈
C of indecomposable modules of the intermediate series over L as follows: They all have basis
{xn,yn|n ∈1
2Z} such that (in all cases, c acts trivially, and n ∈ Z, r ∈1
2+ Z, q,k ∈1
2Z)
Aa,b: Lnxk= (a − k + bn)xn+k, Lnyk=?a − k + (b +1
Trxk= −2(b + 1)xk+r,
Gqxk= yq+k,
2)n?yn+k,
Tryk= −(2b + 1)yk+r,
Gqyk= (−1)2q(a − k + 2bq + q)xq+k,
(1.3)
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and
Ba,b: Lnxk= (a − k + bn)xn+k,Lnyk=
? ?a − k + (b −1
?a − k + (b +1
?
(2b + 1)ryk+r, if k ∈1
if q + k ∈ Z,
if q + k ∈1
if k ∈ Z,
if k ∈1
2)n?yn+k if k ∈ Z,
if k ∈ Z,
2+ Z,
2)n?yn+k
if k ∈1
2+ Z,
Trxk= xk+r,Tryk=
0
Gqxk=
?
?
(−1)2q(a − k + 2bq)yk+q
(−1)2q+1yk+q
xk+q
2+ Z,
Gqyk=
−(a − k + 2bq + q)xk+q
2+ Z,
(1.4)
The modules A0,−1, A0,−1
2have deformations denoted by A1(α), A2(α), α ∈ C, defined by
A1(α) :Lnxk=
?
−(k + n)xn+k
−n(n + α)xn+k
0
if k ∈1
if k = 0,
2Z∗,
Lnyk= −(k +n
2)yn+k,
Trxk=
?
?
if k ∈1
if k = 0,
2Z∗,
−2rxr+k
Tryk= yk+r,
Gqxk=
yq+k
if k ∈1
2Z∗,
(2p + α)yq+k if k = 0,
Gqyk= (−1)2q+1(k + q)xq+k,
(1.5)
A2(α) :Lnxk= −(k +n
2)xn+k,Lnyk=
?
−kyn+k
n(n + α)y0 if k = −n,
0 if k ∈1
2ry0 if k = −r,
yq+k
(2p + α)y0 if k = −q.
if k ∈1
2Z\{−n},
Trxk= −xk+r,Tryk=
?
?
2Z\{−r},
Gqyk= (−1)2q+1kxq+k, Gqxk=
if k ∈1
2Z\{−q},
(1.6)
Similarly, the modules B0,−1
2, B 1
2,−1
2have deformations denoted by B1(α), B2(α),
B1(α) :Lnxk= −(k +n
2)xn+k, Lnyk=


?



−(k + n)yn+k
−n(n + α)yn+k if k = 0,
−kyn+k
0 if k ∈1
−2yr+k if k = 0,
if q + k ∈ Z,
if q + k ∈1
if k ∈ Z∗,
(2p + α)xq+k if k = 0,
kxk+q
if k ∈1
if k ∈ Z∗,
if k ∈1
2Z∗,
2+ Z,
Trxk= xk+r,Tryk=
Gqxk=
?
(−1)2q+1(k + q)yk+q
(−1)2q+1yk+q
xq+k
2+ Z,
Gqyk=





2+ Z,
(1.7)
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B2(α) : Lnxk= (1
2− k −1
2n)xn+k, Lnyk=



?




(1
2− k − n)yn+kif k ∈ Z,
(1
2− k)yn+k
n(n + α)yn+k
if k ∈1
if k =1
2+ (Z\{−n}),
2− n,
Trxk= xk+r,Tryk=
0 if k ∈1
2Z\{1
2− r,
2− r},
2yr+k if k =1
Gqxk=


?





(−1)2q(1
2− k − q)yk+q
(−1)2q+1yk+q
(−1)2q+1(2q + α)yk+q
xk+q
(k −1
if q + k ∈ Z,
if q + k ∈1
if k =1
2+ (Z\{−q}),
2− q,
Gqyk=
if k ∈ Z,
if k ∈1
2)xk+q
2+ Z.
(1.8)
The main result can be formulated as the following theorem.
Theorem 1.2. Any indecomposable module of the intermediate series V over the twisted
N = 2 superconformal algebra is one of the modules Aa,b, Ba,b, A1(α), A2(α), B1(α), B2(α),
or one of their quotient modules for some a,b,α ∈ C.
§2. The indecomposable Harish-Chandra module Aa,b
It is well known that a module of the intermediate series over the super-Virasoro algebra
NS is one of the three series of the modules Aa,b, A(α), B(β) or their quotient modules for
suitable a,b,α,β ∈ C, where Aa,b, A(α) have basis {xk|k ∈ Z}∪{yu|u ∈1
has basis {xu|u ∈1
Aa,b: Lnxk= (a − k + bn)xn+k, Lnyu=?a − u + n(b +1
Gpxk= yk+p,
2+Z} and B(β)
2+Z}∪{yk|k ∈ Z} such that c acts trivially and (see, e.g., [12, 14, 15])
2)?yn+u,
2)?xp+u,
Gpyu= −?a − u + 2p(b +1
A(α) : Lnxk= −(k + n)xn+k, k ?= 0, Lnx0= −n(n + α)xn, Lnyu= −(u +n
Gpxk= yk+p, k ?= 0, Gpx0= (2p + α)yp, Gpyu= (u + p)xp+u,
B(β) : Lnxu= −(u +n
Gpxu= yp+u, u ?= −p, Gpx−p= (2p + β)y0, Gpyk= kxp+k,
for n,k ∈ Z, p,u ∈1
Let V = V0⊕ V1be any indecomposable L-module with dimVλ
Z2, where
2)yn+u,
2)xn+u,Lnyk= −kyn+k, k ?= −n, Lny−n= n(n + β)y0,
2+ Z.
α? 1 for all λ ∈ H∗, α ∈
Vλ
α= {v ∈ Vα| L0· v = λ(L0)v}.
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It is easy to see that c acts trivially on V (see, e.g., [7, 9]). So we can omit c in the following.
Being indecomposable, it is clear that there exists a ∈ C such that
V =?
⊕
k∈1
2Z
Va+k
0
?⊕?
⊕
k∈1
2Z
Va+k
1
?. (2.1)
Obviously V′= ⊕k∈ZVa+k
the intermediate series over NS. First we assume that they are modules of type Aa,b(later
on, we shall consider all possible deformations). Thus we can choose a basis {xk,yk|k ∈1
of V such that V0is spanned by {xi,yi+1
with the following relations, for some a, b, b′∈ C,
0
⊕⊕k∈1
2+ZVa+k
1
and V′′= ⊕k∈1
2+ZVa+k
0
⊕⊕k∈ZVa+k
1
are modules of
2Z}
2|i ∈ Z} and V1is spanned by {xi+1
2,yi|i ∈ Z},
Lmxi=
?
(a − i + bm)xm+i
(a − i + b′m)xm+i if i ∈1
? ?a − j + m(b′+1
?a − j + m(b +1
Gpxk= yk+p,
?
−?a − i + 2p(b +1
2+ Z, i,j,k ∈1
if i ∈ Z,
2+ Z,
Lmyj=
2)?ym+j
if j ∈ Z,
if j ∈1
2)?ym+j
2+ Z,
Gpyi=
−?a − i + 2p(b′+1
2)?xp+i if i ∈ Z,
2)?xp+i
if i ∈1
2+ Z,
(2.2)
where m ∈ Z, r,p ∈1
2Z. To determine possible structures on V , we suppose
Trxk= fr,kxk+r,Tryk= f′
r,kyk+r,Gnxk= gn,kyk+n,Gnyk= g′
n,kxk+n,
(2.3)
for n ∈ Z, r ∈
defined in (2.2) and (2.3) by Aa,b,b′ temporarily.
1
2+ Z, k ∈
1
2Z and some fr,k, f′
r,k, gn,k, g′
n,k∈ C. Denote such a module
Lemma 2.1. (i) If there is some r0∈1
fixed n ∈ Z, there is always infinitely many m ∈ Z satisfying fr0+n,k0+m?= 0. The analogous
result also holds for f′
2+ Z, k0∈1
2Z such that fr0,k0?= 0, then for any
r,k,gn,kand g′
n,k.
(ii) For any r0 ∈
fr0+n,k0+m?= 0, or there are infinitely many pairs (n,m) ∈ Z × Z with f′
(iii) For any n0 ∈ Z, k0 ∈
gn0+n,k0+m?= 0. Similar result also holds for g′
1
2+ Z, k0 ∈
1
2Z, there are infinitely many pairs (n,m) ∈ Z × Z with
r0+n,k0+m?= 0.
1
2Z, there are infinitely many pairs (n,m) ∈ Z × Z with
n,k.
Proof. We shall prove (i) for gn,k. The rest can be proved similarly. Suppose there is some
n0,k0∈ Z with gn0,k0?= 0. For m,n,k ∈ Z, applying [Lm,Gn] = (m
2− n)Gm+nto xkgives
gn,k
?a − n − k + m(b′+1
2)?− (a − k + bm)gn,m+k= (m
5
2− n)gm+n,k.(2.4)