We investigate the Atiyah-Singer index theorems with torsion given by Neveu-Schwarz three-form flux H under the condition d H = 0 in flux compactification scenarios with non-trivial background fields in string theories. Using an identification between the Clifford algebra on the geometry and the canonical quantization condition in [Formula: see text] quantum mechanics, we explicitly reformulate the Dirac index on manifolds with torsion, which will provides a fundamental information to effective theories derived from string theory. In the same analogy we also reformulate the Euler characteristics and the Hirzebruch signatures in the framework of [Formula: see text] quantum mechanics.
INDEX THEOREMS ON TORSIONAL GEOMETRIES
