Vanishing vacuum amplitude of four-dimensional heterotic string theory compactified on N=2 superconformal field theory

We study a (2,2) compactification of the heterotic string theory by forming a tensor product of nine copies of level-1, N=2 minimal superconformal series. We obtain a generalized Riemann θ identity and show that the one-loop vacuum amplitude vanishes, as is expected from N=1 space-time supersymmetry.