Explicit formula for singular vectors of the Virasoro algebra with central charge less than 1

We calculate explicitly the singular vectors of the Virasoro algebra with the central charge c ≤ 1. As a result, we have an infinite sequence of the singular vectors for each Fock space with given central charge and highest weight, and all its elements can be written in terms of the Jack symmetric functions with rectangular Young diagram. Since the paper of Belavin, Polyakov and Zamolodchikov [1], singular vectors of the Virasoro algebra attract a lot of attention because of its relationship with correlation functions. Many attempts were done to make the explicit expression of these vectors in the Verma module (see for example [2-4]), however it remains to be unclear what is the structure of these vectors. On the other hand, singular vectors on the Fock space are known to have relationships with symmetric functions. Goldstone, Wakimoto and H. Yamada showed that when the central charge c is equal to 1, then the singular vectors of smallest degree are proportional to the Schur symmetric functions with rectangular Young diagram [5, 6]. Using the actions of the Virasoro generators on the Schur symmetric 1 functions, [7] generalized this result to the case c < 1 and expressed the singular vectors as a sum of the Schur symmetric functions. Mimachi and Y. Yamada extended these results to the general value of c and showed that the singular vector of the Fock space FAr+1,s+1 (see below for definition) with degree rs is proportional to the Jack symmetric function with rectangular Young diagram (s) [8, 9]. However, as is well known, there are infinitely many singular vectors in each Fock space FAr+1,s+1 if c ≤ 1, and all above results determine only the singular vectors of smallest degree. Recently, [10] demonstrated that the set of all the bosonized Jack symmetric functions forms a natural basis of the Fock space, and conjectured explicitly the actions of the Virasoro generators on this Jack basis. In this paper, applying these results, we determine the explicit expression of all the singular vectors of the Fock space FAr+1,s+1 when central charge is less than 1. As a result, we found that all the singular vectors are proportional to the Jack symmetric functions with rectangular Young diagram, which indicates further deep relationships between the Jack symmetric functions and the representations of the Virasoro algebra. The set-up is as follows. Denote the Virasoro generators as usual by Ln (n ∈ Z) and the central charge as c, with commutation relations [Ln, Lm] = (n−m)Ln+m + n(n − 1) 12 c δn+m,0. (1) As a representation space, we take the Fock space FA defined by FA := C[a−1, a−2, a−3, · · · ]|A〉, (2) where we use the bosonic operators [an, am] = n δn+m,0 (n,m ∈ Z), and a vacuum vector |A〉 defined by a0|A〉 = A|A〉, an|A〉 = 0 (n ∈ Z>0) (3) Then the bosonic representation of Ln’s is given as follows — the Feigin-Fuchs representation [11], Ln = 1 2 ∑ k∈Z : an−kak : −A1,1(n+ 1)an, (4)