EXPLICIT FORMULA FOR VIRASORO SINGULAR VECTOR
DMITRY V. MILLIONSHCHIKOV
We denote by V ir the Virasoro algebra with generators {C, Li , i ∈ Z} and V (h, c) stands
for a Verma module over V ir, v is its vacuum vector. A nontrivial vector w ∈ V (h, c) is called
singular, if Li w=0 for all positive i. There is a singular vector w with grading n in a Verma
module V (h, c) if and only if there exist two positive integers p and q and a complex number
t 6= 0 such that pq = n and
1 − p2
1 − pq 1 − q 2 −1
c = c(t) = 13 + 6t + 6t−1 , h = hp,q (t) =
t+
+
t .
4
2
4
For fixed positive integers p and q and a fixed complex number t the Verma module V (hp,q (t), c(t))
containes a homogeneous singular vector wp,q (t) of degree pq defined uniquely up to multiplication
by some scalar:
X
i1 ,...,is
(t)L−i1 . . . L−is v,
wp,q (t) = Sp,q (t)v =
Pp,q
i1 +···+is =pq
1,...,1
(t)
Pp,q
is equal to one.
We assume that the coefficient
In 1988 Benoist and St-Aubin found an explicit expression for the series S1,p (t):
X
(p−1)!2 ts−p
L−i1 . . . L−is ,
(1)
S1,p (t) =
Pl
Qs−1 Pl
q=1 iq )
q=1 iq )(p−
l=1 (
i1 +. . .+is =p
Later Bauer, Di Francesco, Itzykson and Zuber, using the formula (1) as an initial step,
proposed an algorithm for finding all singular vectors. But in practice, their algorithm meets
with serious computational difficulties and it is still unclear how to get with its help an explicit
formula for all Sp,q (t). However, the expressions for S2,2 (t) and S2,3 (t) found by means of it
allowed us to guess the general formula for singular vectors of the entire series S2,p (t).
We prove that for a Verma module V (h, c) over the Virasoro algebra with
1
c=c(t)=13+6t+6t−1 , h=h2,p (t)=− (p−1+t)(t−1 (p+1)+3) , t ∈ C, t 6= 0.
4
we have the following formula for S2,p (t)
X
f2,p (t; i1 , . . . , is )L−i1 . . . L−is ,
(2)
S2,p (t) =
i1 +. . .+is =2p
.
where the sums are over all partitions of 2p by positive numbers without any ordering restrictions
and the coefficients f2,p (t; i1 , . . . , is ) are defined by the formulas
2p−1
s
m−1
Q
Q
P
(p−t−r)
(2t−1)(im −1)+2p−1−2
in
(2p−1)!2 (2t)s−2p
r=1
m=1
n=1
(3)
f2,p (t; i1 ,. . .,is )=
.
l
2p−1
s−1
l
l
Q
Q
P
P
P
in )(2p−
in )(p−t−
in )
(2p−1−2k)
(
k=0
l=1
n=1
n=1
n=1
Lomonosov Moscow State University, Department of Mechanics and Mathematics, Leninskie
gory, 1, Moscow, 119992, Russia
E-mail address: [email protected]
This work is supported by the Russian Science Foundation under the grant 14-11-00414 at Steklov
Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.
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