NO GHOST THEOREM AND COHOMOLOGY THEOREM FOR STRINGS IN ARBITRARY STATIC BACKGROUNDS

We consider a string moving in an arbitrary time-independent background given by an arbitrary conformal field theory of appropriate central charge (e.g., c = 25 for bosonic string) and one flat time-like dimension. We show that the physical subspace of the Hilbert space is positive semi-definite (no ghost theorem) and that the cohomology of the BRST operator is trivial except for the ghost number one (for open bosonic string) sector (cohomology theorem). Both the proofs are reductio ad absurdum proofs based on the corresponding theorems for the strings moving in flat background. In cases where there is an extra flat space-like dimension (besides the flat time-like one), the transverse subspace with positive-definite norm can be constructed.