A note on the extended superconformal algebras associated with manifolds of exceptional holonomy

It was observed some time ago by Shatashvili and Vafa that superstring compactification on manifolds of exceptional holonomy gives rise to superconformal field theories with extended chiral algebras. In their paper, free field realisations are given of these extended superconformal algebras inspired by Joyce's constructions of such manifolds as desingularised toroidal orbifolds. The purpose of this note is to give another realisation of these algebras starting not from free fields, but from the superconformal algebras associated to Calabi--Yau manifolds. These superconformal algebras, originally studied by Odake, are extensions of the N=2 Virasoro algebra. For the case of G_2 holonomy, our realisation is inspired in the conjectured construction of such manifolds as a desingularisation of (K x S^1)/Z_2, where K is a Calabi--Yau 3-fold admitting an antiholomorphic involution. Similarly, for the case of Spin(7) holonomy our realisation suggests a construction of such manifolds as desingularisations of K'/Z_2, where K' is a Calabi-Yau 4-fold admitting an antiholomorphic involution.