CFT description of string theory compactified on non-compact manifolds with G2 holonomy

We construct modular invariant partition functions for strings propagating on non-compact manifolds of G2 holonomy. Our amplitudes involve a pair of N=1 minimal models Mm, Mm+2 (m=3,4,…) and are identified as describing strings on manifolds of G2 holonomy associated with Am−2 type singularity. It turns out that due to theta function identities our amplitudes may be cast into a form which contain tricritical Ising model for any m. This is in accord with the results of Shatashvili and Vafa. We also construct a candidate partition function for string compactified on a non-compact Spin(7) manifold.