Three-generation solutions of equations of motion in heterotic supergravity

We study the generation number of massless fermions in compactifications recently found heterotic supergravity. The internal spaces are products two-dimensional constant curvature, and standard embedding is not assumed. constrained by equations motion Bianchi identity. In case that Euler characteristics three submanifolds $(2,\ensuremath{-}2,\ensuremath{-}2)$, or less. also show three-generation solutions exist for arbitrary negatively curved 2-manifolds, we present some explicit solutions.