CFT Description of String Theory Compactified on Non-compact Manifolds with G₂ Holonomy

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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.