Twisting of N=1 SUSY Gauge Theories and Heterotic Topological Theories

It is shown that $D=4$ $N=1$ SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on compact K\"ahler manifold. The conditions of cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a non-trivial set of physical operators defined as classes of the cohomology of this BRST \op . We prove that the physical correlators are independent on external K\"ahler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a K\"ahler manifold. The correlators of local physical \op s turn out to be independent of anti-holomorphic coordinates defined with a complex structure on the K\"ahler manifold. However a dependence of the correlators on holomorphic coordinates can still remain. For a hyperk\"ahler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical \op s.