  We study various topological invariants on a torsional geometry in the
presence of a totally anti-symmetric torsion H under the closed condition dH =
0, which appears in string theory compactification scenarios. By using the
identification between the Clifford algebra on the geometry and the canonical
quantization condition of fermions in quantum mechanics, we construct N=1
quantum mechanical sigma model in the Hamiltonian formalism. We extend this
model to N=2 system, equipped with the totally anti-symmetric tensor associated
with the torsion on the target space geometry. Next we construct transition
elements in the Lagrangian path integral formalism and apply them to the
analyses of the Witten indices in supersymmetric systems. We explicitly show
the formulation of the Dirac index on the torsional manifold which has already
been studied. We also formulate the Euler characteristic and the Hirzebruch
signature on the torsional manifold.
