Direct proofs of the Feigin-Fuchs character formula for unitary representations of the Virasoro algebra

TL;DR

This paper provides a direct, elementary proof of the Feigin-Fuchs character formula for unitary Virasoro algebra representations at the critical central charge c=1, extending previous results for 0<c<1.

Contribution

It introduces a new direct proof method for the Feigin-Fuchs character formula at c=1 using primary fields and irreducibility of Virasoro actions, complementing existing approaches.

Findings

Confirmed irreducibility of Virasoro actions on multiplicity spaces at c=1

Demonstrated that singular vectors are given by Goldstone's formulas

Provided an alternative proof using Jantzen filtration and Kac determinant formula

Abstract

Previously we gave a proof of the Feigin--Fuchs character formula for the irreducible unitary discrete series of the Virasoro algebra with 0<c<1. The proof showed directly that the mutliplicity space arising in the coset construction of Goddard, Kent and Olive was irreducible, using the elementary part of the unitarity criterion of Friedan, Qiu and Shenker, giving restrictions on h for c=1-6/m(m+1) with m>2. In this paper we consider the same problem in the limiting case of the coset construction c=1. Using primary fields, we directly establish that the Virasoro algebra acts irreducibly on the multiplicity spaces of irreducible representations of SU(2) in the two level one irreducible representations of the corresponding affine Kac--Moody algebra. This gives a direct proof that the only singular vectors in these representations are those given by Goldstone's formulas, which also play an important part in the proof. For this proof, the theory is developed from scratch in a self-contained semi-expository way. Using the Jantzen filtration and the Kac determinant formula, we give an additional independent proof for the case c=1 which generalises to the case 0<c<1, where it provides an alternative approach to that of Astashkevich.