A note on mirror symmetry for manifolds with spin(7) holonomy

Starting from the extended worldsheet superconformal algebras associated with G2 manifolds, we extend the algebra to Joyce's orbifolds with spin(7) holonomy. We show how the mirror symmetry in manifolds with spin(7) holonomy arises as the automorphism in the extended sperconformal algebra. In one class of Joyce's orbifolds, the automorphism of the superconformal algebra can be realized as 14 kinds of T-dualities along the supersymmetric T4 toroidal fibrations in the manifolds with spin(7) holonomy. In this class of examples, Joyce's orbifolds are pairwise identified under the mirror symmetry. We then discuss some interesting features of the mirror symmetry on the manifolds with exceptional holonomy and how it is different from the Calabi-Yau mirror symmetry.